In this work we study cat maps with many degrees of freedom. Classical cat maps are classified using the Cayley parametrization of symplectic matrices and the closely associated centre and chord generating functions. Particular attention is dedicated to loxodromic and elliptic-hyperbolic behaviour, which are new features of four-dimensional maps: we construct a map that is not Anosov, but is ergodic and mixing. The maps are then quantized using a Weyl representation on the torus and the general condition on the Floquet angles is derived for a particular map to be quantizable. The semiclassical approximation is exact, regardless of the dimensionality or of the nature of the fixed points. We single out the study of the quantum period function (QPF), that is the period of the quantum map as a function of the finite Hilbert space dimension. It is found that the QPF depends basically on the ergodicity and on the existence of degenerate Lyapunov exponents for some power of the cat map.